Electron Paramagnetic Resonance:Hyperfine Interactions

Chem 634 T. Hughbanks

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Reading✿ Drago’s Physical Methods for Chemists is still a good

text for this section; it’s available by download (zipped, password protected): http://www.chem.tamu.edu/rgroup/dunbar/chem634.htm. For EPR: Chapters 9 and 13.

✿ Also good: Inorganic Electronic Structure and Spectroscopy, Vol. 1, Chapter 2 by Bencini and Gatteschi; and Vol. 2, Chapter 1 by Solomon and Hanson.

✿ Background references: § Orton, Electron Paramagnetic Resonance; § Abragam & Bleaney Electron Paramagnetic Resonance of

Transition Ions; § Carrington & McLachlan, Introduction to Magnetic Resonance; § Wertz & Bolton, Electron Spin Resonance.

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What is EPR (ESR)?

• Electron Paramagnetic (Spin) Resonance • Applies to atoms and molecules with one or more unpaired

electrons • An applied magnetic field induces Zeeman splittings in

spin states, and energy is absorbed from radiation when the frequency meets the resonance condition, hν = ∆E ∝ µB × H

• 1/λ ~ 1 cm–1, ν = c/λ ~ 1010 s–1 (microwave, GHz)

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Spectroscopy: The Big PictureEPR

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What information do we get from EPR?

• Chemists are mostly interested in two main pieces of information: g-values and hyperfine couplings, though spin-relaxation information can also be important to more advanced practitioners.

• g-values: these are, in general, the structure-dependent (i.e., direction-dependent) ‘proportionality constants’ that relate the electron spin resonance energy to the direction of the applied magnetic field: hν = gx,y,zµBH. The direction-dependence is determined by electronic structure.

• Hyperfine coupling: arise from interactions between magnetic nuclei and the electron spin and give information about the delocalization of the unpaired

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What background do we need?

• We need an understanding of the magnetic properties of atoms and/or open-shell ions. In particular, we will need to understand the role of spin-orbit coupling, which is essential for understanding g-value anisotropy.

• Quantum mechanical tools: manipulation of angular momentum operators and understanding of how perturbation theory works.

• Basic ideas from MO theory and ligand field theory must be familiar.

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Experimental Aspects• EPR performed at fixed frequency (i.e., field is

varied). Some common frequencies: ü X-band: ν ~ 9.5 GHz, (9.5 × 109 s–1; λ ~ 3 cm);

typical required field is ~ 3400 G = 0.34 T - most common (hν/geµB = 3389.94 G for ν = 9.500 GHz)

K-band: ν ~ 24 GHz (2.4 × 1010 s–1; λ ~ 1 cm); typical required field is ~ 8600 G = 0.86 T - less common (hν/geµB = 8564.06 G for ν = 24.00 GHz)

ü Q-band : ν ~ 35 GHz (3.5 × 1010 s–1; λ ~ 0.8 cm); typical required field is ~ 12500 G = 1.25 T (hν/geµB = 12489.25 G for ν = 35.00 GHz)

• Solvents: generally avoid water, alcohols, high dielectric constants (why?)

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Experimental Aspects• Form of samples:

– Gases, liquids, solids, crystals, frozen solutions – Glasses are best for solution samples measured

below solvent freezing points because they give homogeneity, whereas solutions have cracks and frozen crystallites and scatter radiation

• Note: glasses can form with pure solvents or mixtures. Unsymmetrical molecules have a greater tendency to form glasses. (e.g., cyclohexane crystallizes, methylcyclohexane tends to form glasses). H-bonding promotes crystallization.

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Experimental Aspects• Containers:

– Pure silica (“quartz”), not Pyrex (borosilicate glass); the latter absorbs microwaves to too great an extent

• Constant, homogeneous magnetic fields. • Microwave generator (klystron) - sets up a standing

wave, for which the frequency is fixed by the geometry of the cavity.

• To improve the S/N ratio (reduce the effect of 1/f noise), the field is generally modulated (100-200 kHz) and detected with a lock-in detector (amplifier). Phase-sensitive detection gives a true differential of line shapes. For this reason, EPR spectra are normally displayed as derivatives – this is actually convenient since it accentuates features of the absorption in which we are interested anyway.

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The Experiment• EPR:

– Fixed microwave frequency, sweep the field

– In NMR, frequencies are varied at fixed fields

– The power absorbed is measured as a function of field:

Modulated Field H(t) = Hocos(ωt); where H=field t = some time constant and cos(ωt) = modulation frequency

N Shν

KLYSTRONMICROWAVEGENERATOR

MODULATEDPOWERSUPPLY

MAGNET

The field is modulated to improve the S/N ratio

STANDINGMICROWAVE

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H-atom Spin Hamiltonian

• 1st term: electronic Zeeman • 2nd term: nuclear Zeeman • 3rd term: Fermi Contact hyperfine (isotropic)

– magnitude depends on the electron density of the unpaired electron on the nucleus, ψ (0).

H Spin = geµBH •S− gNµNH • I + aS • I

a = 8π3geµBgNµN ψ (0)

2

ge = 2.002322... ; gH = 5.585486...

µB =e!

2mec= 9.2731×10−21 erg/G

9.2731×10−24 J/T

⎧⎨⎪

⎩⎪

µN = e!2mpc

= 5.0504×10−24 erg/G

5.0504×10−27 J/T

⎧⎨⎪

⎩⎪*

* This is the “nuclear magneton” - a constant that is not ‘nucleus specific’ and is not the nuclear moment symbol used by Drago.

a = 1420.4058 MHz = 9.411708×10−18 erg

9.411708×10−25 J

⎧⎨⎪

⎩⎪ψ (0)

2= 2.14813 electrons/Å3

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Electronic Zeeman interaction for H-atom

If a hydrogen atom is placed in a magnetic field, H = Hz z, the electron’s energy will depend on its mS value. The Zeeman interaction between the applied field and the magnetic moment of the electron is illustrated as:

ˆ

H

geµBH

ge = 2.002322… mS = −1 2

mS = 1 2

Nuclear spin neglected!

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H-atom Spin Energies

|αeαN〉|αeβN〉

|βeβN〉|βeαN〉

|αeαN〉|αeβN〉

|βeβN〉|βeαN〉

Electronic Zeeman

Nuclear Zeeman 1st-order

hyperfine

– geµBH12

geµBH12

|αe〉

|βe〉

geµBH•S

gNµNH12

– gNµNH12

– gNµNH12

gNµNH12

–gNµNH•I

a14

– a14

|αeαN〉

|βeβN〉

|αeβN〉

|βeαN〉

– a14

a14

+aS•I

2nd-order hyperfine

turns on mixing!

a(Sx Ix + Sy I y ) =12 a(S+ I− + S− I+ )

aSz Iz

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H-atom Spectrum

H

hv1 = hv2

E(αeαN )− E(βeαN ) = E(αeβN )− E(βeβN )

12geµBH1 −

12gNµNH1 +

a4

⎛⎝⎜

⎞⎠⎟− − 1

2geµBH1 −

12gNµNH1 −

a4

⎛⎝⎜

⎞⎠⎟

= 12geµBH2 +

12gNµNH2 −

a4

⎛⎝⎜

⎞⎠⎟− − 1

2geµBH2 +

12gNµNH2 +

a4

⎛⎝⎜

⎞⎠⎟

(neglects 2nd -order hyperfine)

geµBH1 +a2= geµBH2 −

a2

∴H1 − H2 !ageµB

= 506.82 G

~ 1420 MHz

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Zero-field H-atomaS i I = a(Sx Ix + Sy I y + Sz Iz ) = a( 1

2 (S+ I− + S− I+ )+ Sz Iz )

aS i I α eα N = a 12( ) 1

2( ) α eα N = 14a α eα N

aS i I βeβN = a − 12( ) − 1

2( ) βeβN = 14a βeβN

aS i I 12

α eβN ± βeα N( )⎡⎣

⎤⎦ = a( 1

2 (S+ I− + S− I+ )+ Sz Iz )12

α eβN ± βeα N( )⎡⎣

⎤⎦

= 12 a(S+ I− + S− I+ ) 1

2α eβN ± βeα N( )⎡

⎣⎤⎦ + a

12( ) − 1

2( ) 12

α eβN ± βeα N( )⎡⎣

⎤⎦

± 12 a

12

βeα N ± α eβN( )⎡⎣

⎤⎦ −

14a 1

2α eβN ± βeα N( )⎡

⎣⎤⎦

for + : = 14a 1

2α eβN + βeα N( )⎡

⎣⎤⎦ for −: = − 3

4a 1

2α eβN − βeα N( )⎡

⎣⎤⎦

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Exact H-atom Spin Hamiltonian

α eα N α eβN βeα N βeβN

α eα N

α eβNβeα N

βeβN

12 Δe − ΔN( )+ 1

4 a − E 0 0 0

0 12 Δe + ΔN( )− 1

4 a − E 12 a 0

0 12 a − 1

2 Δe + ΔN( )− 14 a − E 0

0 0 0 − 12 Δe − ΔN( )+ 1

4 a − E

⎡

⎣

⎢⎢⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥⎥⎥

where Δe = geµBH and ΔN = gNµN HSolutions:

E = + 14 a ± 1

2 Δe − ΔN( )

E = − 14 a ±

a2

⎛⎝⎜

⎞⎠⎟

2

+Δe + ΔN

2⎛⎝⎜

⎞⎠⎟

2

from: Carrington & McLachlan, Introduction to Magnetic Resonance16

H-atom EPR; variable field

from: Carrington & McLachlan, Introduction to Magnetic Resonance17

H-atom EPR; variable field

from: Carrington & McLachlan, Introduction to Magnetic Resonance18

most EPR measurements are

from: Carrington & McLachlan, Introduction to Magnetic Resonance19

A Note on units of energy• Three different units of energy (none of them really are

energy !) are used in EPR: • cm–1 (1/λ), GHz or Hz or MHz (ν), and gauss (G). The conversions are done as follows: • cm–1 × c (2.99793 × 1010 cm/s) → Hz or cm–1 × 29.9793 → GHz • Hz × h/geµB (3.5683 × 10-7 G-s) → G or MHz × 0.35683 → G or cm–1 × 10697 → G The conversion to gauss yields the field necessary to

induce an equivalent Zeeman splitting for a free electron. Example: a zero field splitting parameter of 0.1012 cm–1 could

also be reported as 3.034 GHz or 1082 G.

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Methyl radical splitting diagram

• The four transitions for the methyl radical. +mI states are lowest for ms = –1/2 and the –mI state lowest for ms = 1/2, from the I•S term.

From Drago, “Physical Methods…” - Fig. 9-7 – corrected.

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Methyl radical splitting diagram

• The four transitions for the methyl radical. +mI states are lowest for ms = –1/2 and the –mI state lowest for ms = 1/2, from the I•S term.

Plotted to reflect the constant-frequency experimental conditions.

H

E

MI = +3/2

MI= +1/2

MI = –1/2

MI= –3/2

MI = +3/2

MI= +1/2

MI = –1/2

MI= –3/2

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Methyl radical Spectrum

• The four transitions for the methyl radical. +mI states are lowest for ms = –1/2 and the –mI state lowest for ms = –1/2, from the I•S term.

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Organic π Radicals

–

–

O

O

–

aH = QρH ; Q ! 22.5 G

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ESR Spectrum: Naphthalide Anion–

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ESR Spectrum: Naphthalide Anion

aβ = 4.94 MHz

aα = 13.79 MHz~ 4.92 G

~ 1.76 G

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Naphthalide Anion - SOMO

B2g-0.258 (-0.263)

0.429 (0.425)

Numbers are coefficients from ESR (Hückel)

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Nuclei with I > 1/2 • A nucleus with spin I

splits the electron resonance into 2I + 1 peaks.

• Again, the electronic Zeeman interaction is much larger than the hyperfine interaction.

N

O

N

O

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[VO(H2O)5]2+: V4+, d1, S = 1/2; 51V(100%), I = 7/2

• Characteristic vanadyl EPR • g-tensor is only modestly anisotropic (g = 1.9658)

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Nitronylnitroxide; Two 14N For n identical nuclei,

hyperfine splitting yields 2nI + 1 lines.

N

N

O

O

Ph

N

N

O

O

Ph

H hyperfine = aS • (I1 + I2 )

(I1, I2)

(1,−1)(0,0)(−1,1)

(1,0) (−1,0)(0,1) (0,−1)

(1,1) (−1,−1)

⎧

⎨

⎪⎪⎪

⎩

⎪⎪⎪

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Delocalized Electrons in Clusters

• 59Co: 100% abundance

• I = 7/2 • 2(3 × 7/2)+1 = 22 expected lines

Co

Co

Co CO

CO

C

OC

OC

COCO

OCO OC

Se

SOMO (a2)

Single crystal, 77K, H || z32

Anisotropic hyperfine coupling

Simulated EPR spectrum of a normal copper complex, tetragonal Cu(H2O)6

2+ (at X-band, n = 9.50 GHz). (A) EPR absorption; (B) first derivative spectrum without and (C) with copper hyperfine splitting.

63Cu 69.1% I = 3/2 A0= 4.95 GHz 65Cu 30.9% I = 3/2 A0= 5.30 GHz

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55Mn (100%)Mn2+ doped into MgV2O6 gx = 2.0042 ± 0.0005 gy = 2.0092 ± 0.001 gz = 2.0005 ± 0.0005 Dx = 218±5 G; Dy = -87±5 G; Dz = -306±20 G; from H. N. Ng and C. Calvo, Can.

J. Chem. 50, 3619 (1972).Excited 4T2g state mixes into 6A1g

Assign all the transitions with initial and final values of MS and mI!

From Drago (corrected)

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Multiple hyperfine splittings

O

NH1 H2

O

CuIINH1H2

63Cu (I = 3/2) 100% enriched 14N (I = 1)1H (I = 1/2)

A gorgeous case of accidental degeneracy:

4 multiplets; each multiplet has 11 lines with relative intensity ratio:

1:2:3:4:5:6:5:4:3:2:1EPR external standard, DPPH (diphenyl-picrylhydrazide), g = 2.0037 ± 0.0002

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Organic Radicals in Solids

• 1st term: electronic Zeeman (g-tensor anisotropy neglected)

• 2nd term (S • T • I): nuclear Zeeman • 3rd term: Hyperfine – includes the isotropic

Fermi contact term and the anisotropic, dipolar, part of the hyperfine (S • T′ • I):

H Spin = geµBH • S − gNµN H • I + S • T • I

S • T • I = S • ′T • I + aS • I

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Dipolar Hyperfine Interaction

• The energy of dipole-dipole interactions falls off as the cube of the distance between dipoles - whether the dipoles are electric or magnetic.

r̂ =

rr

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Dipolar Hyperfine Interaction• The electron is distributed over a volume defined by

an orbital while the nucleus can be regarded as fixed at a point. Thus, the dipolar interaction is summed (integrated) over the electron’s probability distribution, |ψ |2.

ψ

µe • µN

r3−

3(µe • r)(µN • r)

r5ψ

The classical energy of interaction between two magnetic dipoles, µe and µN , is

Edipolar =µe • µN

r3−

3(µe • r)(µN • r)

r5~

1r3

where r is the vector separating the dipoles. The corresponding QM Hamiltonian is

H dipolar = −geµBgNµNS • Ir3

−3(S • r)(I • r)

r5

⎡

⎣⎢

⎤

⎦⎥

|ψ |2

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Hyperfine Anisotropy: (S•T′)•I dominant over nuclear Zeeman, H•I

If the dipolar hyperfine interaction dominated over the externally applied field (it often doesn’t), it would determine the axis of quantization of the nucleus. The hyperfine tensor determines the net angle between electron’s axis (along S) and the nucleus’s axis, along S•T′.

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Dipolar Hyperfine: Origin of 2nd-order transitions

When dipolar hyperfine interaction is comparable with the Zeeman field (a common case), it would determine the axis of quantization of the nucleus changes depending on whether electron is spin-up or spin-down. The hyperfine tensor determine the net angle between electron’s axis (along S) and the nucleus’s axis, along S•T′.

First-order EPR transitions we've seen involve the flipping of only the electron spin,

αeβN ↔ βeβN and αeαN ↔ βeαNThis is so because αN βN = 0 and αN αN = βN βN = 1

If the axis of quantization changes, both spins can flip

′αN ′′βN2= sin2 θ

2 and ′αN ′′αN

2= cos2 θ

2 Therefore, for some orientations, satellite transitions are

observed. Detailed study yields signs of the tensor components.

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Hyperfine Anisotropy

• Matrix elements of T′ should be interpreted as expectation values of the operators shown over the electronic wavefunction. (The origin is at the nucleus at which the spin I resides.)

H dipolar = −geµBgNµNS • ′T • I

T = ′T + a1

′T =

r2 − 3x2

r5

−3xyr5

−3xzr5

−3xyr5

r2 − 3y2

r5

−3yzr5

−3xz

r5

−3yz

r5

r2 − 3z2

r5

⎡

⎣

⎢⎢⎢⎢⎢⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥⎥⎥⎥⎥⎥

The classical energy of interaction between two magnetic dipoles, µe and µN , is

Edipolar =µe • µN

r3−

3(µe • r)(µN • r)

r5~

1r3

where r is the vector separating the dipoles. The corresponding QM Hamiltonian is

H dipolar = −geµBgNµNS • Ir3

−3(S • r)(I • r)

r5

⎡

⎣⎢

⎤

⎦⎥

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Hyperfine Anisotropy

• Matrix elements of T′ should be interpreted as expectation values of the operators shown over the electronic wavefunction. (The origin is at the nucleus at which the spin I resides.)

The classical energy of interaction between two magnetic dipoles, µe and µN , is

Edipolar =µe • µN

r3−

3(µe • r)(µN • r)

r5~

1r3

where r is the vector separating the dipoles. The corresponding QM Hamiltonian is

H dipolar = −geµBgNµNS • Ir3

−3(S • r)(I • r)

r5

⎡

⎣⎢

⎤

⎦⎥

3cos2θ −1

θ − π 2 = sin−1 1 3 = 35.2644°

θ = cos−1 1 3 = 54.7356°

θ − π 2

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Hyperfine Anisotropy; Example: Malonate Radical

• The malonate radical is obtained by irradiation of malonic acid with x-rays. The trapped radical can be studied with single-crystal EPR - which reveals the anisotropy of the hyperfine tensor, T′. (We can initially ignore the Zeeman interaction of the proton, H•I, i.e., consider only the first-order effects resides.)

HHHOOC

HOOCH≡

OO

H

O

H

OH O

O

H

OH

H

O

≡

OO

O

H

OH

H Hhν(x-ray)

SOMO

See Carrington & McLachlan, pp. 103-7.43

H-field orientations for single-crystal EPR experiment

H ⊥ to molecular plane

H ! to molecular plane, ⊥to π orbital & C–H bond

H ! to molecular plane & C–H bond, ⊥to π orbital

HOOC

HOOCH

C = – 61

A = – 29

B = – 91

Principal axes of T and observed hyperfine splittings (MHz)

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Hyperfine Anisotropy; Example: [VO(CN)5]3–

• The EPR spectrum of the [VO(CN)5]3– ion in a single crystal of KBr is shown above; the magnetic field is aligned along the [100] axis of the KBr host.

a) What are the values of A|| and A⊥? (Hint: There are two overlapping spectra shown here. Think about the way in which the [VO(CN)5]3– ion is likely to substitute for K+ and Br– ions in the host; the relative intensities of the two spectra should settle which spectrum corresponds to the || direction and which corresponds to the ⊥ direction.)

See Wertz & Bolton, p. 320.

VC

O

CC

C

C

NN

NN

N3–

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[VO(CN)5]3–, cont.

(b) Explain the relative magnitudes of the hyperfine constants.

(c) Identify all the lines in the spectrum. g|| = 1.9711; what is the value of g⊥?

(d) Draw a d-orbital splitting diagram for this ion (π-effects are important too) and use the g-value information to determine as many of the d-orbital energy splittings as you can from the information you have so far (for V4+, ζ = 248 cm–1) assuming the orbitals have pure d-character.

(e) Consider the effects of covalence in the calculation of the g-values for this ion and discuss how the important ligand bonding effects should influence both g|| and g⊥.

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